Characterization of exact two-query quantum algorithms

Quantum query model is a crucial model for quantum computing, where one query to some input variable of a Boolean function f defined on ${\{0,1\}}^n$ returns the variable value. The exact query complexity, denoted as $Q_E\left(f\right)$, is defined to be the minimum number of queries required to determine the function value. An important problem in this area is to give a succinct characterization of a k-query exact quantum algorithm for an arbitrary k. To date, the cases $k=1$ and $k=n$ are already solved and the case $k=2$ remains unknown. Our result is that there are 27 nondegenerate Boolean functions up to isomorphism with $Q_E\left(f\right)$ being two, among which only two functions can be solved by a 2-query classical algorithm. The input bit number n of the above 27 functions ranges from 2 to 6, where the case $n\le 3$ is already proved and the case $n=4$ is already found by numerically solving semidefinite programming, which is a complete characterization of quantum query algorithm. Assuming the correctness of the numerical result for $n=4$, we prove that there are four functions in the case $n=5$, one in the case $n=6$ and none in the case $n\ge 7$. We further show that the 25 functions for which quantum algorithm has advantage over classical algorithm contain essentially only four different structures.